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Computers and Concrete, Vol. 3, No. 5 (2006) 313-334 313Design optimization of reinforced concrete structuresAndres Guerra†and Panos D. Kiousis‡Colorado School of Mines, Division of Engineering, 1500 Illinois St, Golden, CO. 80401, USA(Received April 18, 2006, Accepted August 25, 2006)Abstract.A novel formulation aiming to achieve optimal design of reinforced concrete (RC) structuresis presented here. Optimal sizing and reinforcing for beam and column members in multi-bay and multi-story RC structures incorporates optimal stiffness correlation among all structural members and results incost savings over typical-practice design solutions. A Nonlinear Programming algorithm searches for aminimum cost solution that satisfies ACI 2005 code requirements for axial and flexural loads. Materialand labor costs for forming and placing concrete and steel are incorporated as a function of member sizeusing RS Means 2005 cost data. Successful implementation demonstrates the abilities and performance ofMATLAB’s (The Mathworks, Inc.) Sequential Quadratic Programming algorithm for the design optimization ofRC structures. A number of examples are presented that demonstrate the ability of this formulation toachieve optimal designs.Keywords: sequential quadratic programming; cost savings; reinforced concrete; optimal stiffness distri-bution; optimal member sizing; RS means; nonlinear programming; design optimization.1. IntroductionThis paper presents a novel optimization approach for the design of reinforced concrete (RC)structures. Optimal sizing and reinforcing for beam and column members in multi-bay and multi-story RC structures incorporates optimal stiffness correlation among structural members and resultsin cost savings over typical state-of-the-practice design solutions. The design procedures for RCstructures that are typically adapted in practice begin by assuming initial stiffness for the structuralskeleton elements. This is necessary to calculate the internal forces of a statically indeterminatestructure. The final member dimensions are then designed to resist the internal forces that are theresult of the assumed stiffness distribution. This creates a situation where the internal forces usedfor design may be inconsistent with the internal forces that correspond to the final designdimensions. The redistribution of forces in statically indeterminate structures at incipient failure,however, results in the structural performance that is consistent with the design strength of eachmember. Although this common practice typically produces safe structural designs, it includes aninconsistency between the elastic tendencies and the ultimate strength of the structure. In somecases this can cause unsafe structural performance under overloads (e.g. earthquakes) as well asunwanted cracking under normal building operations when factored design loads are close to serviceloads, (e.g. dead load dominated structures). This inconsistency also implies that such designs areunnecessarily expensive as they do not optimize the structural resistance and often result in† Graduate Student, E-mail: aguerra@mines.edu‡ Associate Professor, Corresponding Author, E-mail: pkiousis@mines.edu314 Andres Guerra and Panos D. Kiousismembers with dimensions and reinforcement decided by minimum code requirements rather thanultimate strength of allowable deflections.Because of its significance in the industry, optimization of concrete structures has been the subjectof multiple earlier studies. Whereas an exhaustive literature review on the subject is outside thescope of this paper, some notable optimization studies are briefly noted here. For example, Ballingand Yao (1997), and Moharrami and Grierson (1993) employed nonlinear programming (NLP)techniques for RC frames that search for continuous-valued solutions for beam, column, and shearwall members, which at the end are rounded to realistic magnitudes. In more recent studies, Lee andAhn (2003), and Camp, et al. (2003) implemented Genetic Algorithms (GA) that search fordiscrete-valued solutions of beam and column members in RC frames. The search for discrete-valued solutions in GA is difficult because of the large number of combinations of possible memberdimensions in the design of RC structures. The difficulties in NLP techniques arise from the need toround continuous-valued solutions to constructible solutions. Also, NLP techniques can becomputationally expensive for large models.In general, most studies on optimization of RC structures, whether based on discrete- orcontinuous-valued searches, have found success with small RC structures using reduced structuralmodels and rather simple cost functions. Issues such as the dependence of material and labor costson member sizes have been mostly ignored. Also, in an effort to reduce the size of the problems,simplifying assumptions about the number of distinct member sizes have often been made based onpast practices. While economical solutions in RC structures typically require designs where groupsof structural elements with similar functionality have similar dimensions, the optimal characteristicsand population of these groups should be determined using optimization techniques rather thanpredefined restrictions. These issues are addressed, although not exhaustively, in this paper, byincorporating more realistic costs and relaxed restrictions on member geometries.This study implements an algorithm that is capable of producing cost-optimum designs of RCstructures based on realistic cost data for materials, forming, and labor, while, at the same time,meeting all ACI 318-05 code and design performance requirements. The optimization formulation of theRC structure is developed so that it can be solved using commercial mathematical software such asMATLAB by Mathworks, Inc. More specifically, a sequential quadratic programming (SQP)algorithm is employed, which searches for continuous valued optimal solutions, which are roundedto discrete, constructible design values. Whereas the algorithm is inherently based on continuousvariables, discrete adaptations relating the width and reinforcement of each element are imposedduring the search.This optimization formulation is demonstrated with the use of design examples that study thestiffness distribution effects on optimal span lengths of portal frames, optimal number of supportsfor a given span, and optimal sizing in multi-story structures. RS Means Concrete and MasonryCost data (2005) are incorporated to capture realistic, member size dependent costs.2. Optimization2.1. RC structure optimizationThe goal of optimization is to find the best solution among a set of candidate solutions usingefficient quantitative methods. In this framework, decision variables represent the quantities to beDesign optimization of reinforced concrete structures 315determined, and a set of decision variable values constitutes a candidate solution. An objectivefunction, which is either maximized or minimized, expresses the goal, or performance criterion, interms of the decision variables. The set of allowable solutions, and hence, the objective functionvalue, is restricted by constraints that govern the system.Consider a two dimensional reinforced concrete frame with i members of length Li. Each memberhas a rectangular cross section with width biand depth hi, which is reinforced with compressive andtensile steel reinforcing bars, and respectively (Fig. 1). The set of bi, hi, , andconstitute the decision variables. The overall cost attributed to concrete materials, reinforcing steel,formwork, and labor is the objective function. The ACI-318-05 code requirements for safety andserviceability, as well as other performance requirements set by the owner, constitute the constraints.The formulation of the problem and the associated notation follow:Indices:i: RC structural member; beam or column.m: steel reinforcing bar sizes.Sets:Columns:set of all members that are columns.Beams:set of all members that are beams.Sym:set of pairs of column members that are horizontally symmetrically located on thesame story level.Horiz:same as Columns, but activated only when the structure is subjected to horizontalloading.et:set of member types; either Columns or Beams.Parameters:Cconc, mat’l= 121.00 $/m3- Material Cost of ConcreteCsteel(et) = 2420 $/metric ton for beam members and 2340 $/metric ton for column membersLi- Length of member i, meters (typically 4 to 10 meters)d' = 7 cm = Concrete Cover to the centroid of the compressive steel - same as the cover to thecentroid of thee tensile steel.= 28 MPa - Concrete Compressive Strengthβ1= 0.85 - Reduction Factor = 28 MPaEc = 24,900 MPa - Concrete Modulus of ElasticityAs1 i,As2 i,As1 i,As2 i,f

c′f

c′Fig. 1 Reinforced concrete cross section and resistive forces316 Andres Guerra and Panos D. KiousisEs = 200,000 MPa - Reinforcing Bar Modulus of Elasticityfy= 420 MPa - Steel Yield Stress= Stress in Tensile Steel≤fybar_numbering = Metric equivalent bar sizes = [#13, #16, #19, #22, #25]bar_diamm= Rebar diameters for m = 1:5. i.e., [12.7, 15.9, 19.1, 22.2, 25.4] mmbar_aream= Rebar areas for m = 1:5. i.e., [129, 199, 284, 387, 510] mm2= 0.01 - Minimum ratio of steel to concrete cross - sectional area in all column members= 0.08 - Maximum ratio of steel to concrete cross - sectional area in all column members= 0.0033 - Minimum ratio of steel to concrete cross - sectional area in all beam membersDecision Variables:Primary Variables:bi- width of member i (cm)hi- depth of member i (cm)As1,i- Compressive steel area of member i (cm2)As2,i- Tensile steel area of member i (cm2)Auxiliary Variables:pi- Perimeter of member i, 2*(bi+ hi) for columns, and (bi+ 2*hi) for beamsCforming(bi, hi) - Cost of forms in placce ($/SMCA) as a function of cross-sectional area as described inFig. 2.1Cconc, place(bi, hi) - Cost of placing concrete ($/m3) as a function of corss-sectional area as described inFig. 2.2Pui- Factored Internal Axial Force of member i determined via FEA (kN)Mui- Factored Internal Moment Force of member i determined via FEA (kN · m)ci- Distance from most compressive concrete fiber to the neutral axis for member i (cm)- Location of the plastic centroid of member i (cm) from the most compressive fiber.Formulation:(1)subject to:(2)(3)(4)(5)(6)(7)f

s′ρmincρmaxcρminbxiC( ): minpiLiCformingbihi,( ) +⋅ ⋅bihiAs1 i,As2 i,⋅–⋅( ) LiCconc mat'l,Cconc place,bihi,( )+( ) +⋅ ⋅As1 i,As2 i,+( ) LiCsteelet( )⋅ ⋅i 1=n∑bihi= i∀ Columns∈bibj= i j≠( )∀ Sym∈hihj= i j≠( )∀ Sym∈As1 i,As1 j,= i j≠( ) Sym∈∀As2 i,As2 j,= i j≠( ) Sym∈∀As1 i,As2 i,= i Horiz∈∀Design optimization of reinforced concrete structures 317(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)The objective function C, in Eq. (1), describes the cost of a reinforced concrete structure andincludes, in order of appearance, forms in place cost, concrete materials cost, concrete placementand vibrating including labor and equipment cost, and reinforcement in place using A615 Grade 60steel including accessories and labor cost. The costs of forming and placing concrete are a functionof the cross-sectional dimensions b and h of the structural elements. These costs are detailed inTable 1 and in Figs. 2, 3, and 4. As shown in Figs. 2 through 4, linear interpolation between pointsis used to calculate cost of forming and the cost of placing concrete. Note that RS Means providesonly the discrete points. The assumption of linear interpolation between these points is made by theauthors due to lack of better estimates.The constraints in Eqs. (2) through (7) define relative geometries for members in one of thespecified sets: Columns, Sym, and Horiz. Eq. (8) defines the location of the plastic centroid ofelement i as a function of the decision variables. Eq. (9), defines the location of the neutral axis. Eq.xi0.85 bihifc′hi2---- As1 i,fyd′ As2 i,fyhid′–( )⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅0.85 bihifc′As1 i,fyAs2 i,fy⋅+⋅ ⋅ ⋅ ⋅ ⋅------------------------------------------------------------------------------------------------------------- i∀=ci0.003hid′–0.003 fyEs⁄+------------------------------- i∀=Pui0.8 φi0.85fc′bihiAs1 i,As2 i,+( )–⋅( ) fyAs1 i,As2 i,+( ) i∀⋅+⋅ ⋅ ⋅≤As1 i,As2 i,≤ i∀bihi≤ i∀hi5 bi⋅≤ i∀bi3 6 8⁄ bar_diamMAs2ibar_areaM⁄( ) 2 +⁄⋅+ +≤As2 i,bar_areaM⁄( ) 2 1–⁄( ) max 1.0 1.0 bar_diamM⋅,( ) i∀⋅ρmin i,bAs1 i,bihi⋅-----------As2 i,bihi⋅-----------+≤ i∀ Beams∈As2 i,bihi⋅----------- 0.0206fs′fy----As1 i,bihi⋅-----------+≤ i Beams∈∀ρmin i,cAs1 i,bihi⋅-----------As2 i,bihi⋅-----------+≤ i∀ Columns∈As1 i,bihi⋅-----------As2 i,bihi⋅----------- ρmax i,c≤+ i∀ Columns∈Muiφ⁄ ao– a1– Puiφ⁄ a2Puiφ⁄( )2– a3Puiφ⁄( )3a4Puiφ⁄( )4– a5Puiφ⁄( )50≤––bi16 cm i∀≥ hi16 cm i∀≥ As1 i,258 mm2i∀≥ As2 i,258 mm2i,∀≥,,,bi500 cm i∀ hi500 mm i∀≤ As1 i,130 000 mm2i∀,≤ As2 i,130 000 mm2i∀,≤,,,≤318 Andres Guerra and Panos D. Kiousis(10) ensures that the applied factored axial load Puis less thanφPnfor the minimum requiredeccentricity, as defined by ACI 318-05 Eq. (10-2). Eq. (11) maintains that the tensile steel area isgreater than the compressive steel area. The intent of this restriction is to facilitate the algorithmicsearch. Eqs. (12) and (13) are problem specific restrictions related to the width, bi, and depth, hi, ofall members. Whereas these restrictions are common practice in low seismicity areas, they are by nomeans general requirements for all construction. While Eq. (12) ensures that the width is less thanthe depth, Eq. (13) prevents the creation of large shear walls and maintains mostly frame action forthe design convenience of this study. Eq. (14) ensures that the tensile steel can be placed in elementi with appropriate spacing and concrete cover as specified by ACI 318-05. In Eq. (14), the subscriptM on bar_diam and bar_area corresponds to the discrete bar area that is closest to and not less thanthe continuous value of As2,i. The constraints listed in Eqs. (15) through (18) ensure that the amountof reinforcing steel is between code specified minimum and maximum values. And finally, Eq. (19)ensures that the applied axial and bending forces of element i, Puiand Mui, determined with a FiniteElement Analysis (FEA), are within the bounds of the factored P-M interaction diagram which isTable 1 RS Means 2005 concrete cost data all data in english units “from means concrete & masonry cost data 2005.Copyright Reed Construction Data, Kingston, MA 781-585-7880; All rights reserved.”Product Description Total Cost Incl.Overhead and ProfitUnitsREINFORCING IN PLACE A615 Grade 60,including access. LaborBeams and Girders, #3 to #7 2420 (2200) $/metric ton ($/ton)Columns, #3 to #7 2340 (2125) $/metric ton ($/ton)CONCRETE READY MIX Normal weight4000 psi 121.0 (92.5) $/m3($/Yd3)PLACING CONCRETE and Vibrating, including labor andequipment.Beams, elevated, small beams, pumped.(small =< 929 cm2 (144 in2))79.8 (61.0) $/m3($/Yd3)Beams, elevated, large beams, pumped.(large =>929 cm2 (144 in2))53.0 (40.5) $/m3($/Yd3)Columns, square or round, 30.5 cm (12") thick, pumped 79.8 (61.0) $/m3($/Yd3)45.7 cm (18") thick, pumped 53.0 (40.5) $/m3($/Yd3)70.0 cm (24") thick, pumped 51.7 (39.5) $/m3($/Yd3)91.4 cm (36") thick, pumped 34.0 (26.0) $/m3($/Yd3)FORMS IN PLACE, BEAMS AND GIRDERSInterior beam, job-built plywood, 30.5 cm (12") wide, 1 use 41.0 (12.5) $/SMCA* ($/SFCA)70.0 cm (24") wide, 1 use 35.8 (10.9) $/SMCA ($/SFCA)Job-built plywood, 20.3 × 20.3 cm (8" × 8") columns, 1 use 41.0 (12.5) $/SMCA ($/SFCA)30.5 × 30.5 cm (12" × 12") columns, 1 use 37.1 (11.3) $/SMCA ($/SFCA)40.6 × 40.6 cm (16" × 16") columns, 1 use 36.3 (11.05) $/SMCA ($/SFCA)70.0 × 70.0 cm (24" × 24") columns, 1 use 36.7 (11.2) $/SMCA ($/SFCA)91.4 × 91.4 cm (36" × 36") columns, 1 use 34.3 (10.45) $/SMCA ($/SFCA)*Square Meter Contact Area and Square Foot Contact AreaDesign optimization of reinforced concrete structures 319modeled as a spline interpolation of five strategically selected (Mn, Pn) pairs. The lower and upperbounds designate the range of permissible values for the decision variables. The lower bounds onthe width and depth are formulated from the code required minimum amount of steel and theminimum cover and spacing. Upper bounds decrease the range of feasible solutions by excludingexcessively large members.2.2. Optimization technique for RC structuresVarious optimization algorithms can be used depending on the mathematical structure of theproblem. MathWork’s MATLAB is used to apply an SQP optimization algorithm to the describedproblem through MATLAB’s intrinsic function “fmincon”, which is designed to solve problems ofthe form:Find a minimum of a constrained nonlinear multivariable function,f(x),subject toFig. 2 Cost of FORMS IN PLACE for columnsFig. 3 Cost of FORMS IN PLACE for beams320 Andres Guerra and Panos D. Kiousiswhere x are the decision variables, g(x) andh(x)are constraint functions, f(x) is a nonlinearobjective function that returns a scalar (cost), andIband ub are the lower and upper bounds on thedecision variables. All variables in the optimization model must be continuous.The SQP method approximates the problem as a quadratic function with linear constraints within eachiteration, in order to determine the search direction and distance to travel (Edgar and Himmelblau 1998).g x( ) 0;=h x( ) 0;≤Ib x ub;≤ ≤Fig. 4 Cost of PLACING CONCRETE and vibrating, including labor and equipmentFig. 5 Optimization routine flow chartDesign optimization of reinforced concrete structures 321The flow chart in Fig. 5 demonstrates the entire optimization procedure from generating initialdecision variable values, xo, to selecting the best locally optimal solution from a set of optimalsolutions found by varying xo. Initial decision variable values are found by solving the describedoptimization formulation for each individual element subjected to internal forces of an assumedstiffness distribution. At least ten different assumed stiffness distributions are utilized; each leads toa local optimal solution. Comparison of all local optimal solutions, not all of which are different,provides a reasonable estimation of the global optimum solution. Whereas the initial decision isbased on an element-by-element optimization approach, the final optimization (Eqs. 1-20) is globaland allows all element dimensions to vary simultaneously and independently in order to achieve theoptimal solution. As such, the final design is achieved at an optimal internal stiffness configuration.This corresponds to the internal force distribution that ultimately results in the most economicaldesign.3. Design requirements3.1. Cross-section resistive strengthConsider a concrete cross section reinforced as shown in Fig. 1, subjected to axial loading andbending about the z-axis. The resistive forces of the RC cross-section include the compressivestrength of concrete and the compressive and tensile forces of steel, and are calculated in termsof the design variables (b, h, As1,i, As2,i), the location of the neutral axis c, and the concrete andsteel material properties. It is assumed that concrete crushes in compression at εc=0.003 and thatthe strains associated with axial loading and bending very linearly along the depth of the cross-section. The bending resistive capacity Mnfor a given compressive load Pnis calculatediteratively by assuming εc=0.003 at the most compressive fiber of the cross-section, and byvarying c until force equilibrium is achieved. The strength reduction factor is calculated based onFig. 6 Interaction diagram at failure state322 Andres Guerra and Panos D. Kiousisthe strain in the tensile steel. At this state, the resulting moment is evaluated, and the pair (Mn,Pn) at failure is obtained. The locus of all (Mn, Pn) failure pairs is known as the M-P interactiondiagram for a member (Fig. 6).3.2. M-P interaction diagramSafety of any element i requires that the factored pairs of applied bending moment and axialcompression fall within the M-P interaction diagram. The strength reduction factor,φ, is evaluated based on the strain of the most tensile reinforcement and is 0.65 for tensile strain lessthan 0.002, 0.9 for tensile strains greater than 0.005, and is linearly interpolated between 0.65 and 0.9 forstrains between 0.002 and 0.005, as defined in ACI-318-05, Section 9.3. Finally, an axial compressioncutoff for the cases of small eccentricity was placed equal to as perACI-318-05 Eq. (10-2). Mathematically, if is a function that describes the interactiondiagram, safety requires that for all i members. For a given cross-section, theinteraction diagram is typically obtained point-wise by finding numerous combinations (Mn, Pn) thatdescribe failure. For the purpose of this study, the interaction diagram is modeled as a cubic splinebased on five points (Fig. 6), three of which are the balance failure point , the point ofzero moment, and the point that corresponds to a neutral axis location at the level of thecompressive steel axis. The remaining two points are located either above or below the balancefailure point depending on whether the applied axial compression load is greater orsmaller than Pnb, respectively. Fig. 6 shows the three fixed points as solid circles and the twoconditional points which are located below or above the balance failure point as open circles andopen triangles, respectively.It is assumed that the design for shear loads does not alter the optimal design decision variablesbi, hi, As1,i, and As2,i. This assumption is typically acceptable for long slender elements where thecombinations of flexural and axial loads commonly control the element dimensions. It is alsoassumed that the optimal solution is not sensitive to connection detailing. For structures in SeismicDesign Category A, B, and C as classified in the ASCE 7 Standard (SEI/ASCE 7-98) thisassumption is acceptable.3.3. Rounding the continuous solutionConcrete design is ultimately a discrete design problem, where typical element dimensions aremultiples of 50 mm and steel reinforcement consists of a finite number of commonly availablereinforcing bars. Rounding biand hito discrete values is incorporated through the use of asecondary optimization process that finds the optimal reinforcing steel amounts for fixed biand hi.Various combinations of rounding biand hieither up or down to 5 cm multiples are examined tofind the discrete solution with the lowest total cost.Selecting a discrete number and size of longitudinal reinforcing steel from continuous-valuedsolutions is accomplished by finding the discrete number and size that is closest to and not less thanthe continuous-valued optimal solution. This is implemented into the optimization model so that theminimum width bminthat is required to fit the selected reinforcing steel becomes a lower bound thatensures proper cover and spacing of the longitudinal reinforcing steel. This process begins bycalculating the discrete number of bars that are necessary to provide at least the steel area of thecontinuous solution. The minimum width required for proper cover and spacing for each set ofMuiφ⁄ Puiφ⁄,( )Puφ⁄ 0.8 0.85fc′AgAst–( ) fyAst+[ ]=F MnPn,( ) 0=F Muiφ⁄ Puiφ⁄,( ) 0≤MnbPnb,( )MnbPnb,( )Design optimization of reinforced concrete structures 323reinforcing steel bars is calculated. Next, the combined cost of each bar set and the correspondingminimum required width is evaluated. The required width that is associated with the lowestcombined cost is used as the minimum width requirement.It is noted that when a discrete solution requires more than five reinforcing bars, the minimumrequired width is calculated based on bundles of two bars placed side-by-side. Also, for beammembers, the compression and tensile reinforcements are designed to use the same size bars forconstruction convenience, as long as the strength requirements are still met. Finally, stirrups consistof #13 reinforcing bars.4. Design examples4.1. Optimization design examplesThree examples of optimal design for multi-story and multi-bay reinforced concrete frames arepresented here to demonstrate the method. The first example studies the optimal design of a one-story portal frame with varying span length. The second example studies the optimal design of amulti-bay one-story frame with varying number of bays but with a constant over-all girder length of24 meters. The third example creates optimal designs of multi-story, single-bay RC structures withand without horizontal seismic forces.Designs based on standard approaches are created to compare optimal and typical-practicedesigns. While the same cost function used in the optimization formulation is implemented tocalculate the Typical Design Costs, the design dimensions for the Typical Design Costs are based ona simplified state-of-the-practice design method. This method initially assumes that all columns willbe 25 cm by 25 cm and that the width of all beams is 25 cm. Additionally, for beams the depth isequal to the width times a factor of one-third the beam length in meters (McCormac 2001). At thispoint an internal stiffness distribution has been defined and internal forces can be calculated. Theamount of reinforcement is then designed to meet strength and code requirements for each member.The width of each beam member, or width and depth of each column member, is increased inincrements of 5 cm if the assumed size of the member is not sufficient to hold the needed steel tomaintain strength requirements. Note that the internal forces are based on the initially assumedstiffness distribution and not on the design dimensions.For all examples presented here, the length of columns is four meters, the compressive strength ofconcrete is 28 MPa, the yield stress of steel is 420 MPa, the cover of compressive and tensilereinforcement is 7 centimeters, the unit weight of steel reinforced concrete is 24 kN/m3, and theassociated materials and construction costs are listed in Table 1.4.2. Loading conditionsAll frames examined here are loaded by their self weight wG, an additional gravity dead load,wD= 30 kN/m, and a gravity live load, wL= 30 kN/m.Seismic horizontal forces, wherever they are applied, are determined using the ASCE 7 (SEI/ASCE7-98) equivalent lateral load procedure for a structure in Denver, Colorado that is classified asa substantial public hazard due to occupancy and use, and is founded on site class C soil (SEI/ASCE7-98). A base shear force V is calculated using gravity loads equal to 1.0(wD) + 0.25wL, and is324 Andres Guerra and Panos D. Kiousisthen distributed appropriately to each frame story. Table 2 details the equivalent seismic horizontalforces for the multi-story design examples.The frames that are subjected to gravity loads only are designed for factored loads of 1.2 wG+1.2wD+ 1.6 wL= 1.2 wG+ 84 kN/m. The frames that are subjected to gravity and seismic loads aredesigned for the worse combination of the following factored load combinations (ACI 318-05):1.2 wG+ 1.2 wD+ 1.6 wL= 1.2 wG+84 kN/m1.2 wG+ 1.2 wD+ 1.0 wL+ 1.0 E = 1.2 wG+ 66 kN/m + 1.0 E0.9 wG+ 0.9 wD+ 1.0 E = 0.9 wG+ 27 kN/m + 1.0 ETable 2 Calculated equivalent lateral loads for multi-story, one-bay structuresNumber ofStories:Lateral load at:First floor Second Floor Third Floor Fourth Floor Fifth Floor(kN) (kN) (kN) (kN) (kN)One Story 83.23 -- -- -- --Two Story 54.99 109.98 -- -- --Three Story 41.24 82.48 123.73 -- --Four Story 32.99 65.986 98.981 131.97 --Five Story 27.494 54.99 82.484 109.98 137.47Fig. 7 Portal frameTable 3 Optimal portal frame costs for various span lengthsSpan Length(L)TypicalDesign CostsOptimal DesignCostCost Savings OverTyp. Design CostOptimal Cost perFoot of StructureOptimal Cost perFoot of Beammeters Dollars Dollars Percent Dollars Dollars4 2204.7 1913 13.2 159 4786 3203.8 2979 7.0 213 4978 4547 4504.7 0.9 282 56310 6402.5 6336.8 1.0 352 63412 8559.1 8407.9 1.8 420 70114 11629 10702 8.0 486 76416 14533 13192 9.2 550 82524 33787 27975 17.2 874 1166Design optimization of reinforced concrete structures 3254.3.1. One-story portal frame with varying span lengthConsider a single-story portal frame subjected to gravity factored live and dead loads, wU, totaling84 kN/m. The structural model is shown in Fig. 7. The height of the structure is 4 meters, and thespan, L, varies in order to study the effect of the span length on the optimal solution. Comparison ofthe costs at the specified span lengths is presented in Table 3, while a graphical presentation of thesame data is shown in Fig. 8. The normalized cost per foot is presented in Fig. 9 to demonstrate thepure cost burden per foot associated with the larger span lengths. It should be noted that every pointin the cost calculation of Figs. 8 and 9, with the exception of the typical design points, represents anoptimal solution for the specific structure. Although the cost increases smoothly as a function ofspan length, the associated element cross-sectional dimensions do not follow an equally smoothchange pattern. For spans of length up to 12 meters optimal solutions result in small columns and alarge girder. The girders under such design develop moment diagrams that have small negative end-moments, large positive mid-span moments, and behave almost as simply supported beams. Forspan lengths of 14 m or larger, the pattern changes abruptly to one where columns becomeFig. 8 Optimal costs for varying span lengths in a one-story structureFig. 9 Normalized optimal cost for varying span lengths in a one-story structure326 Andres Guerra and Panos D. Kiousiscomparable in size to the girder, and are associated to bending moment distributions where thegirders have equal negative and positive moment magnitudes. Fig. 10 illustrates the girder momentdiagrams at the optimal solution values for the 12 and 14 meter span lengths. The characteristicsand performance of the one-story portal frame for the 12 and 14 meter span lengths are presented inTables 4 and 5. Puand Muare the critical internal forces of each element at the optimal solution andt is the total computation time in minutes using a Pentium 4 2.20 GHz laptop with a Windows XPPro operating system.Optimal designs result in cost savings of just under 1% for the 8-meter span to 17% for the 24-meter span (See Fig. 8 and Table 3). It is very interesting to note that for certain span lengths thetypical design costs are relatively close to the optimal design costs. This demonstrates the regionswhere the typical practice assumptions result in efficient structures and where they do not. A beamlength of eight meters for a portal frame with four meter long columns appears to be the mostefficient structure for the typical practice assumptions as it results in a cost closest to the optimalcost.4.3.2. Increasing the number of bays in a constant 24-meter spanIn this example, a series of multi-bay one story frames are designed with a total span of 24 meters(Fig. 11). The frames range from a one bay structure supported by two columns (bay length of 24.0m) to a twenty-four bay structure supported by 25 equally spaced columns (bay length of 1.0 m). Afactored gravity load of 1.2 wG+ 84 kN/m is placed on the entire 24-meter girder. The cost of eachFig. 10 Girder moments at optimal solution valuesTable 4 Portal frame, L = 12 meters, cost = $8408t = 3.4 minutes b h As1As2PuMuφElement Number cm cm Comp.Tens.kN kN-m1 30 35 6#25 6#25 575.3 287.3 0.892 30 35 6#25 6#25 575.3 287.3 0.893 40 105 2#25 9#25 107.7 1438.6 0.90Table 5 Portal frame, L = 14 meters, cost = $10702t = 1.7 minutes b h As1As2PuMuφElement Number cm cm Comp.Tens.kN kN-m1 60 65 4#25 13#25 668.2 1374.3 0.902 60 65 4#25 13#25 668.2 1374.3 0.903 40 90 2#25 9#25 511.0 1374.3 0.90Design optimization of reinforced concrete structures 327typical and optimal design is presented in Table 6 and is plotted against the number of bays in Fig.12. This example covers the entire range of combinations in the interaction diagram.The inner girders carry virtually no axial load (tension controlled), while the outer girders areloaded at high eccentricity (tension controlled or transition). The inner columns carry their loadswith very small eccentricity and are controlled by the code cap on axial compression for columns ofsmall eccentricities. Finally, the outer columns are compression controlled with a significant bendingmoment component.Note in Fig. 12, the linear relationship between cost values for frames with 21, 22, and 24 bays.These correspond to solutions where all members are controlled by minimum code requirements formember dimensions and reinforcing steel: 20 cm by 20 cm members with a total of 4#13reinforcing steel bars. For 20 bays or less, minimum code requirements gradually stop controllingthe problem, starting with the outer beams. The lowest cost corresponds to seven spans of 3.4meters each. It is noted again that every design presented in Fig. 12 is optimal for the specificgeometry, i.e., span length.Muφ⁄ Puφ⁄,( )Fig. 11 Frame of length 24 m with n baysTable 6 Optimal cost for increasing number of bays in a 24 meter spanNumber of Bays Typical Design Cost ($) Optimal Design Cost ($) Cost Savings (%)1 -- 27975 --2 -- 14743 --3 11432 10789 5.64 10226 9072 11.35 -- 8443 --6 9692.1 8161 15.87 10048 8117 19.28 -- 8312 --9 10875 8378 23.010 -- 8773 --12 -- 9464 --16 -- 10852 --20 -- 12682 --21 -- 13050 --22 -- 13536 --24 -- 14508 --328 Andres Guerra and Panos D. KiousisIn all cases with 20 bays and less, the model is such that beams increase in size to keep thecolumns as small as possible. Only the two-span and the one-span frames have columns with cross-sectional dimensions larger than 25 cm. In all multi-span structures, the outermost beams carrylarger load and, in most cases, the inner beams are close to minimum values.Substantial costs savings over typical design is demonstrated for multi-bay structures. Note in Fig.12 that after reaching the lowest typical design cost of approximately $9700 for the six-baystructure, the costs increase linearly with each additional bay. The linear increase indicates thatminimum dimensions have been reached for the typical design assumptions. Thus, each additionalbay increases the total cost by the cost of one column and one beam.4.3.3. Multi-story design examplesThe designs presented in this section address three groups of multi-story, single bay frames. Thefirst group consists of frames that have a span length of 4 m, one to six stories and are subjected toFig. 13 Multi-story single-bay structuresFig. 12 Increasing number of bays in a 24 meter spanDesign optimization of reinforced concrete structures 329gravity loads only. The second group is similar to the first group. However, the frames have a spanlength of 10 m. The third group consists of frames that have a span length of 10 m, one to fivestories, and are subjected to gravity and seismic loads.Fig. 13 displays the one, two, and three story frames subjected to gravity and seismic forces.Element numbering and loading for the four-, five- and six-story structures follows the same patternas in the frames presented in Fig. 13. The magnitudes of the seismic forces at each floor are listedin Table 2.4.3.3.1. Multi-story - vertical load onlyGiven the effects of girder length on the optimal design patterns of portal frames, short-span andlong-span, (4m and 10 m respectively) multistory frame designs are examined here. The optimalcosts of multi-story frames subjected to gravity loads only are presented in Fig. 14, where circularmarks represent the long span frames and rectangular marks represent the short span frames. Notethat the long-span results are presented in two groups: open circles for the frames that have an oddnumber of stories (1, 3, or 5), and solid circles for the frames that have an even number of stories(2, 4, or 6). No such distinction is necessary for the short span frames.In general, it can be seen from Fig. 14 that the cost increases linearly with the number of storiesfor both the long- and short-span frames. The linear relation between the number of stories and costis almost exact in the case of short-span frames. On the other hand, a closer examination of thelong-span frames (see data points and their least-square-fit lines) indicates that the odd-story framesare slightly more expensive than the even-story frames.The practical significance of this observation is not clear. Nevertheless, from the theoreticalstandpoint, this is an interesting, and rather unexpected finding that merits further analysis andexplanation. Let us consider the n-story frame of Fig. 15. Note that the end-rotational tendencies ofeach girder are resisted by one column above and one column below at each end, with theFig. 14 Increasing cost of multiple story structures subject to vertical loading only330 Andres Guerra and Panos D. Kiousisexception of the roof girder, where only one column below the girder provides the rotationalresistance at each end of the girder. To assist each of the long girders carry their large moments in acost-effective way, columns tend to be stiff in order to restrict rotation and develop sufficientnegative end moment, which in turn reduces the mid-span positive moment. Thus, considering then-story frame of Fig. 15, the columns Cnunderneath the top girder Gn, tend to be stiff. The bottom-ends of these columns (Cn) also provide large rotational resistance to the next girder Gn-1. As aresult, the next set of columns Cn-1does not need to be as stiff. This is indicated in Fig. 15 by thelabel “soft” next to columns Cn-1. Since the bottom-end of these columns do not provide sufficientrotational stiffness to the next girder Gn-2, the next set of columns Cn-2must be stiff (see label inFig. 15). Thus, an alternating pattern of stiff-soft columns is created. In frames of even number ofstories, this pattern results in an equal number of stiff and soft sets of columns. On the other hand,in frames of odd number of stories, the same pattern results in one extra story of stiff columns.Thus, the odd-story frames are relatively more expensive than the even-story frames. It should bepointed out however, that a soft column at a lower story tends to be stiffer than a soft column at ahigher story, since it carries larger axial load. Tables 7 and 8 detail the optimal solution results forthe four- and five-story frame without lateral loads.For short-span structures, creation of stiff columns to help distribute the moment more efficientlyin the girder is not cost effective, given that girders and columns have a similar length, whichmakes two small columns and one large girder less expensive. Thus in the case of short-spanframes, the stiff-soft pattern described earlier is not efficient, and there is no distinction betweenFig. 15 Column stiffness tendencies for Long-Span multistory frames under gravity loadsDesign optimization of reinforced concrete structures 331odd- and even-story frames.The structural tendencies described above were also observed in the one-story portal framediscussed in section 4.2.1. It was found there that smaller span frames favored small end moments(small columns-larger girder), while the larger span frames were more economical when larger endmoments were developed (large columns-smaller girder). In the one story example of section 4.2.1the transition between “small” and “large” span occurred between 12 m and 14 m. In the multistoryframes of this section the transition occurred at less than 10 m, due to the different end conditionsof the girders.Table 7 Four-story, single bay long span structure, vertical load onlyt = 17.7 minutes b h As1As2PuMuφElement Number cm cm Comp.Tens.kN kN-m1 40 40 4#16 4#16 1834.2 91.3 0.652 40 40 4#16 4#16 1834.2 91.3 0.653 55 55 2#22 14#22 -291.7 739.7 0.904 60 60 4#19 5#25 1371.4 656.9 0.905 60 60 4#19 5#25 1371.4 656.9 0.906 30 75 2#25 6#25 299.6 707.7 0.907 30 30 4#13 4#13 919.6 55.9 0.658 30 30 4#13 4#13 919.6 55.9 0.659 55 55 2#22 14#22 -316.3 727.0 0.9010 55 55 2#19 14#19 456.8 701.0 0.9011 55 55 2#19 14#19 456.8 701.0 0.9012 40 65 2#19 12#19 343.0 701.0 0.90Table 8 Five-story, single bay long span structure, vertical load onlyt = 18.5 minutes b h As1As2PuMuφElement Number cm cm Comp.Tens.kN kN-m1 50 55 2#25 5#25 2374.0 431.5 0.652 50 55 2#25 5#25 2374.0 431.5 0.653 35 70 2#25 7#25 32.7 701.2 0.904 40 40 11#16 11#16 1914.4 269.7 0.655 40 40 11#16 11#16 1914.4 269.7 0.656 50 55 2#22 13#22 -116.1 735.2 0.907 50 50 5#22 15#13 1447.7 491.8 0.898 50 50 5#22 15#13 1447.7 491.8 0.899 50 55 2#22 12#22 117.6 737.9 0.9010 40 40 3#19 10#19 985.0 264.0 0.6511 40 40 3#19 10#19 985.0 264.0 0.6512 45 60 2#22 11#22 -105.2 714.4 0.9013 45 45 5#19 6#25 519.0 480.5 0.6714 45 45 5#19 6#25 519.0 480.5 0.6715 65 95 2#22 6#22 232.7 817.1 0.90332 Andres Guerra and Panos D. KiousisFig. 16 illustrates the same optimal costs for the long-span structures along with typical designcosts. The typical design assumptions resulted in efficient stiffness distributions for the one-storystructure. The two-through six-story structures showed cost savings of 11.6%, 3.0%, 9.4%, 6.9%,and 6.2%, respectively.4.3.3.2. Multi-story frames with lateral loadsThe long-span multi-story frames of the previous section are designed here for gravity and seismicloads, as described in Table 2. Following ACI 318-05 requirements, both heavy (66 kN/m) and light(27 kN/m) gravity loads are considered as calculated in the Section 4.2, on Loading Conditions. Theincreased stiffness requirements due to the lateral loads eliminate the stiff-soft patterns observed inthe gravity-only examples of the previous section. The cost of each optimized design is presented inFig. 17, along with the costs of the gravity only frame designs to indicate the cost increase due tolateral loading. It can be seen that the seismic loads do not cause significant cost burdens for framesFig. 16 Comparison of optimal and typical design costs in multiple story structures subject to vertical loadingonlyFig. 17 Increasing cost of multiple story structures subject to horizontal loadingDesign optimization of reinforced concrete structures 333with three stories or less, but they become fairly costly for taller structures. This observation is sitedependent, and can be different for seismic loads that correspond to a different site.5. ConclusionsThis paper presents a novel approach for optimal sizing and reinforcing multi-bay and multi-storyRC structures incorporating optimal stiffness correlation among structural members. This studyincorporates realistic materials, forming, and labor costs that are based on member dimensions, andimplements a structural model with distinct design variables for each member. The resulting optimaldesigns show costs savings of up to 23% over a typical design method. Comparison betweenoptimal costs and typical design method costs demonstrates instances where typical designassumptions resulted in efficient structures and where they did not. The formulation, including thestructural FEA, the ACI-318-05 member sizing and the cost evaluation, was programmed inMATLAB (Mathworks, Inc.) and was solved to obtain the minimum cost design using the SQPalgorithm implemented in MATLAB’s intrinsic optimization function fmincon.A number of fairly simple structural optimization problems were solved to demonstrate the use ofthe method to achieve optimal designs, as well as to identify characteristics of optimal geometricspacing for these structures.It was found that optimal portal frame designs follow different patterns for small and large baylengths. More specifically it was found that short-span portal frames are optimized with girders thatare stiff compared to the columns, thus ensuring girder simple supported action, while long-spanportal frames are optimized with girders that are approximately as stiff as the columns, thus splittingthe overall girder moment to approximately equal negative and positive parts.It was also found that girders that are supported by multiple supports, as in the case of multi-bayframes, have an optimal span length, below which the design becomes uneconomical because somemembers are controlled by code imposed minimum sizes, and above which the design also becomesuneconomical as the member sizes tend to become excessive.Finally it was found that optimal design multi-story frames present similar characteristics to one-story portal frames where short-bay designs are optimal with girders that are stiff compared to thecolumns, and long-span designs are optimal with girders that are approximately as stiff as thecolumns. For gravity dominated long-span multi-story frames, optimal designs tend to havealternating stiff and soft columns. It is not however clear that this pattern exists in optimal designsof multi-bay multistory frames, considering that the interior columns typically do not carrysignificant moments due to gravity. Finally, the alternating stiff/soft column pattern was notobserved when the horizontal seismic loads had a significant influence on the design.ReferencesAmerican Concrete Institute (ACI)(2002), Committee 318 Building Code Requirements for Structural Concrete(ACI 318-02) and Commentary (ACI 318R-02), Detroit.Balling, R.J. and Yao, X. (1997), “Optimization of reinforced concrete frames”, J. Struct. Eng., ASCE, 123(2),193-202.Camp, C.V., Pezeshk, S., and Hansson, H. (2003), “Flexural design of reinforced concrete frames using a genetic334 Andres Guerra and Panos D. Kiousisalgorithm”, J. Struct. 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